The simplest example of a Teichmüller curve is \((\mathcal {M}_1, \operatorname{Id})\) , where \(\mathcal {M}_1\)  is the moduli space of compact Riemann surfaces of genus one. For \(g \geqslant 2\) , the existence of Teichmüller curves already follows from W. Thurston's construction of pseudo-Anosov mapping classes, see section 8 of [1]}. From [2]} it follows that for any \(g \geqslant 2\)  there exist countably many Teichmüller curves in \(\mathcal {M}_g\) . C. T. McMullen, see [3]}, constructed an infinite and explicit family of (primitive)i.e. These Teichmüller curves don't cover Teichmüller curves in lower genera. Teichmüller curves in \(\mathcal {M}_2\) , he also discovered that the image of any Teichmüller curve in \(\mathcal {M}_2\)  always lies on a Hilbert modular surfaceHilbert modular surfaces, see Hirzebruch's seminal paper [4]} and the book [5]} for a general introduction, live in \(\mathcal {A}_2\) , the moduli space of abelian surfaces. Here we mean the image of the Teichmüller curve under the Torelli map \(\mathcal {M}_2 \rightarrow \mathcal {A}_2\)  always lies in a Hilbert modular surface.. He then extended his construction to genus \(g=3 \,\text{and} \,4\)  in [6]}.
A complete classification of (primitive) Teichmüller curves in \(\mathcal {M}_2\)  follows from the work of [3]}, [8]}, [9]} and [10]}, we refer to section 5.5 in [11]} for more details. W. Veech, in [12]}, constructed two Teichmüller curves for each \(g \geqslant 2\) , the image of both of these lie in the hyperelliptic locus of \(\mathcal {M}_g\) . In [13]}, using algebro-geometric methods a large number of Teichmüller curves are produced, the strong point of this approach being that the family of Riemann surfaces that these Teichmüller curves parametrize can be described explicitly as a (complex) one parameter family of polynomials. For further details on this approach we refer to the excellent article [11]}. More recently, the authors in [15]} have innovated a new description of (certain) Teichmüller curves in terms of theta functions, from their viewpoint these Teichmüller curves lead to a genuinely new type of modular forms.
